Técnicas e instrumentos de recolección de datos

The finite element method can be considered a Rayleigh-Ritz method.

The classical Rayleigh-Ritz technique represents a variational approach whereby a distributed system is approximated by a discrete one by assuming a solution of the differential boundary-value problem as a finite series of admissible functions. Unfortunately, systems with complex geometry or complex boundary conditions cannot be accomodated easily by global admissible functions.

In the finite element method, the approximate solution is constructed using local admissible functions, defined over small subdomains of the structure. Good approximations can be realized with low-degree polynomials. Displacements are calculated by methods based on the principle of virtual work and/or the principle of minimum total potential energy.

Instead of solving differential equations with complicated boundary conditions, the finite element method evaluates integrals of relatively simple polynomial functions. Variational methods put less strict conditions on the functions approximating the displacement field than the analytical methods based on differential equations.

7.1 Principle of virtual work (PVW)

PVW is basically a statement of the static equilibrium of a mechanical system. In the following, the form known as the principle of virtual displacements (PVD) will be used, as applied to elastic bodies.

7.1.1 Virtual displacements

By definition, virtual displacements are:

a) arbitrary (fictitious, virtual);

b) infinitesimal (follow the rules of differential calculus);

c) not related to either the actual displacements or to the forces producing them;

d) continuous in the interior and on the surface of the body;

A continuity C is generally required for bars and elasticity problems, ⁰ while a continuity C is imposed for beams, plates and shells. Exceptions do exist. ¹ Remember that a function of several variables is said to be of class C in a ^m domain V if all its partial derivatives, up to the mth order inclusive, exist and are continuous in the domain V.

e) kinematically admissible, i.e. consistent with the system kinematic boundary conditions (geometric constraints).

If the differential equation of the problem is of order m 2= n, the admissible functions must have continuity Cⁿ⁻¹, i.e. the geometrical boundary conditions must be satisfied to the

(

n 1−

)

th derivative. For bars, m=2 and the assumed functions must have continuity C⁰.

For beams m=4 and the approximating functions must have continuity C1. Because the continuity required is reduced from C² in the governing differential equation to C in the variational equation, the functional is said to have ¹ a “weak form”.

A virtual displacement will be denoted by ‘ δ ’ in front of a letter, e.g. uδ . The symbol ‘ δ ’ was introduced by Lagrange to emphasize the virtual character of the variations, as opposed to the symbol d which designates actual differentials of position coordinates.

Denoting by

{ }

u =_⎣u v w_⎦^T,

the displacement vector (6.4) inside the body or on the surface S_σ with unprescribed displacements (Fig. 6.1), the vector of virtual displacements is

{ }

δu =_⎣δu δv δw_⎦^T. (7.1) The vector of the corresponding virtual strains will be

{ }

δε =

[ ]{ }

B δu . (7.2) where

[ ]

^B is the strain-displacement matrix.

7. ENERGY METHODS 133

7.1.2 Virtual work of external loads

For a bar in tension (Fig. 7.1, a), the virtual work of the external force F is u

F

W_E δ

δ = ⋅ . (7.3)

It has the same value whether the bar material is linear elastic (Fig. 7.1, b) or nonlinear elastic (Fig. 7.1, c). Note that it is simply (force× displacement), because the force is constant along the virtual displacement, the latter being arbitrary, hence independent of the force.

In the general case of loading by conservative body forces (6.1), surface tractions (6.2) and point forces (6.3), the virtual work of external loads is

{ } { } ∫ { } { } ∑ { } { }

∫

^{+ +}

=

i

T i i S

T V

E u T p V u p A u F

W δ d δ d δ

δ

σ

s

v . (7.4)

Note that the scalar product under the first integral is

{ } { }

δu ^T p_v =δu⋅p_vx +δv⋅p_vy +δw⋅p_vz.

a b c Fig. 7.1

Note also the absence of the factor 21 which occurs in the expression of the work of elastic forces, because the external loads remain constant during the action along the virtual displacements.

7.1.3 Virtual work of internal forces

For a three-dimensional continuum, the virtual work of internal stresses is

{ } { }

∫

=

V

I T V

W δ d

δ ε σ (7.5)

as shown in Fig. 7.2 for the uniaxial case.

Fig. 7.2

Again, stresses remain constant during the action on virtual strains.

7.1.4 Principle of virtual displacements

For elastic bodies, the principle of virtual displacements states that:

If a system is in equilibrium, then during an arbitrary small displacement from the equilibrium position, the virtual work of applied loads equals the virtual work of internal forces

I

E W

W δ

δ = . (7.6)

Also: A body is in equilibrium if the internal virtual work equals the external virtual work for every kinematically admissible displacement field.

{ } { }

δ d −

∫ { } { }

δ d −

∫ { } { }

δ d −

∑ { } { }

δ =0

∫

i

T i i S

T V

T V

T V u p V u p A u F

σ

σ

ε _{v s} .

(7.6, a) In (7.6) δW_E is the work of external loads on the virtual displacements

{ }

δu which are independent of loading and kinematically admissible.

If stresses are expressed in terms of a set of parameters defining completely the displacement pattern – the nodal displacements, then equilibrium relations can be obtained and the displacement parameters determined. The nodal displacements do not permit the fully equilibrating position to be reached, so that the PVD will ensure approximate equilibrium.

Note that the virtual work of reaction forces at supports is zero. Since the principle of virtual displacements is an equilibrium requirement, it is independent of material behaviour, i.e. whether the material is elastic or inelastic. It applies only

7. ENERGY METHODS 135

for loading by conservative forces, which do not change direction during the action on the virtual displacements. The external work is independent of the path taken.

Example 7.1

For the three-bar pin-jointed framework shown in Fig. 7.3, loaded by a force F, find the internal bar forces and the displacement of point 4.

Fig. 7.3

Solution. Consider three states of the analyzed system:

1. The initial state, in which bars are not loaded by external forces and are not prestressed (Fig. 7.4, a).

2. The final state of static equilibrium, in which the external force F, of components F₁=Fsinα and F₂ =Fcosα, produces a displacement of the joint 4, of components u and ₁ u (Fig. 7.4, b). ₂

The joint 4 is acted upon by the external forces F , ₁ F and by internal ₂ forces T , ₁ T , ₂ T (Fig. 7.4, c). The joint reacts with forces equal in magnitude but ₃ of opposite sign, producing the elongations Δ₁, Δ₂, Δ₃ (Fig. 7.4, d).

3. An imaginary state, in which the joint 4 is given a virtual displacement of components δu and ₁ δu (Fig. 7.4, e), which produce virtual elongations in bars ₂

δΔ1, δΔ₂, δΔ₃ (Fig. 7.4, f), the applied forces remaining constant.

The virtual displacements δu and ₁ δu and the virtual bar extensions ₂ δΔ₁, δΔ2, δΔ₃ satisfy the compatibility equations (2.19)

. u

u , u

, u

u

cos δ sin δ δ

δ δ

cos δ sin δ δ

2 1

3 2 2

2 1

1

θ θ

Δ Δ

θ θ

Δ

+

−

=

=

+

=

(7.7)

For the three bars, the force-elongation equations (2.21) can be written

1 E1A

=T l

Δ ,

A E

T θ

Δ₂= ^2lcos ,

A E T l3 3=

Δ . (7.8)

Fig. 7.4

Equating internal work to external work (7.6) using the products of real forces and virtual displacements we obtain

2 2 1 1 3 3 2 2 1

1δ T δ T δ F δu F δu

T Δ + Δ + Δ = + . (7.9)

Substituting (7.7) into (7.9) and collecting coefficients of δu and ₁ δu ₂ gives

7. ENERGY METHODS 137

δu₁

(

T₁sinθ−T₃sinθ−F₁

)

+δu₂

(

T₁cosθ+T₂+T₃cosθ−F₂

)

=0. (7.10) Since δu and ₁ δu are unrelated to each other, we could put either to zero. ₂ Equation (7.10) must be zero whatever the values of δu and ₁ δu . This can only ₂ be true if their coefficients vanish

.

These are indeed the equations of equilibrium. It is confirmed that the principle of virtual work is an equivalent statement of statical equilibrium.

Substituting (7.8) in the finite form of (7.7), then in (7.11), the components of the displacement of point 4 can be determined from the following equations

.

7.1.5 Proof that PVD is equivalent to equilibrium equations

Consider a form of equation (7.6, a) without point forces

{ } { } ∫ { } { } ∫ { } { }

where l and m are direction cosines of the outward normal at the surface.

Adding together

( ) [ ( ) ( ) ]

In matrix form (Gauss’ theorem)

∫ { } { }

⁼

∫ { } [ ] { }

⁻

∫ { } [ ] { }

^∂ displacement fields are continuous. This applies only within a single element and up to its surface.

The componets of stresses at element interfaces may not balance at a point, but only in the mean. Most finite elements in use today do not achieve continuous stresses across interfaces.

On substituting in (7.6, b)

{ }

^δ

( [ ] { } { }

^{∂ +}

)

^{d −}

_∫ ^{{ }}

^δ

( [ ] { } { }

⁻

)

^{d =0}

Because

{ }

^δu are arbitrary, its coefficients must vanish. The equations of equilibrium emerge from the brackets

[ ] { } { } { }

∂ ^T σ + p_v = 0 in V,

[ ] { } { } { }

n ^T σ − p_s = 0 on S . _σ

The PVD supplies equilibrium conditions both within and on the surface of the body. So, when using approximate functions for

{ }

u , it is not necessary to worry about the equilibrium boundary conditions. Part of the surface, i.e. S , is _σ supported in some way and there the tractions

{ }

p will be unknown reactions and _s not specified loads. It is conventional to remove the unknown reactions by choosing the virtual displacements

{ }

δu to be zero over S . _σ

7. ENERGY METHODS 139

7.2 Principle of minimum total potential energy

The total potential energy Π of an elastic body is defined as the sum of the strain energy U and the work potential of external loads W _P

WP

U+

Π = . (7.13)

7.2.1 Strain energy

Consider the strain energy (6.32)

{ } [ ]{ }

∫

=

V

T D V

U d

2

1 ε ε .

For a virtual strain

{ }

δε , the virtual increase of strain energy is

{ } [ ]{ } ∫ { } [ ]{ }

∫

⁺

=

V T V

T D V D V

U δ d

2 d 1 2 δ

δ 1 ε ε ε ε .

Because

{ } [ ]{ }

(

^{δε T D ε}

)

^{T =}

^{{ }}

^{ε T}

[ ]{ }

^{D δε ,}

{ } [ ]{ } { } { }

_I

V T V

T D V V W

U δ d δ d δ

δ =

∫

^{ε ε} =

∫

^{ε σ} = ^.

WI

U δ

δ = . (7.14)

For an elastic body, the virtual variation of the strain energy is equal to the virtual work of the internal stresses on virtual strains.

For a bar in tension, substituting σ =Eε and x u d

=d

ε , yields

∫

∫

∫

^{= =}

=

l l

x x A u x E x u

A E V

U

V

d d d d

δ d d

δ d

δ

δ εσ ε ε . (7.15)

For a beam in bending, substituting ²₂ d d y xv

−

ε = (5.4), gives

∫

∫ ∫

_⎟⎟^{⎟ =}

⎠

⎞

⎜⎜

⎜

⎝

⎛

∂

∂

∂

= ∂

l

x x I x E x

A x y

x E U

l A

d d d d

δ d d

δ d

δ ² ₂v ²v₂ ^{2 2} ₂v ²v₂ . (7.16)

The above expressions can be obtained directly from the strain energies

for a bar ⁼

∫

^⎜⎜_⎝^{⎛ ⎟⎟}_⎠^⎞

l

x x u A

U E d

d d 2

2

, (7.17)

for a beam ⁼

∫

^⎜⎜_⎝^{⎛ ⎟⎟}_⎠^⎞

l

x x I

U E d

d d 2

2 2

2v . (7.18)

7.2.2 External potential energy

The work potential of external loads W is equal to the negative product of _P external forces by the corresponding displacement

E

P W

W =− . (7.19)

The negative sign appears because the external loads lose some of their capacity for doing work when displaced in the direction they act.

For example, a gravitational force F=mg acts in the opposite direction to a vertical displacement h and the potential becomes mgh.

An external point load F has potential energy _j

(

−Fjuj

)

instead of

⎟⎠

⎜ ⎞

⎝⎛− F_j u_j 2

1 , because this potential arises from the magnitude of force and its capacity to do work when it moves, being independent of the linear properties of the body on which it acts.

For a three-dimensional continuum

{ } { } ∫ { } { } ∑ { } { }

∫

^{− −}

−

=

i T i i S

T V

P u T p V u p A u F

W

σ

d

d _s

v . (7.20)

For virtual displacements

{ }

^δu

E

P W

W δ

δ =− . (7.21)

7.2.3 Total potential energy

The total potential energy (7.13) can be written

{ } { } ∫ { } { } ∫ { } { } ∑ { } { }

∫

^{− − −}

=

i

T i i S

T V

T V

T V u p V u p A u F

σ

σ ε

Π δ d _v d _s d .

(7.13, a) Its variation is

7. ENERGY METHODS 141

E

P W W

W

U δ δ δ

δ

δΠ = + = _I− . (7.22)

Based on equation (7.6) it follows that 0

δΠ = , (7.22, a)

hence, at equilibrium, the total potential energy has a stationary value. If 0

δ²Π > , the stationary value is a minimum, the equilibrium is stable.

The principle of minimum total potential energy states that:

If a deformable body is in equilibrium under the action of external loads and reaction forces, then the total potential energy has a minimum value.

Reciprocally, if under the action of external loads and reaction forces the total potential energy of a deformable body is a minimum, then it is in a stable equilibrium state.

Thus, it can be considered that (7.22, a) is a condition that establishes or defines the equilibrium, rather than a result of the equilibrium.

An equivalent statement is: For conservative systems, of all possible kinematically admissible displacement fields, the one satisfying equilibrium corresponds to a minimum value of the total potential energy.

Reciprocally, any kinematically admissible displacement field which minimizes the total potential energy represents a stable equilibrium configuration.

Example 7.2

For the truss shown in Fig. 7.3, the strain energy for a bar is

2 1 2

1 ₂

i i i i i

i l

A T E

U = Δ = Δ ,

and the external potential energy is

∑

−

=

i i i

P Fu

W .

Expressing the elongations in terms of displacements, according to the compatibility relations, the total potential energy can be written

( )

(

u u

)

Fu F u .

EA

os u u EA

EA u

2 2 1 2 1 2

1

22 2 2

1

cos 2 sin

c cos 2

2 sin

−

− +

− +

+ +

+

=

θ θ

θ θ θ

Π

l

l l

Cancelling the derivatives of Π with respect to each independent variable 0

1

∂ =

∂ u

Π , 0

2

∂ =

∂ u Π ,

we obtain the equilibrium equations (7.11).

Example 7.3

Apply the principle of minimum total potential energy to a beam in bending, subjected to a distributed load and to the end bending moments and shear forces as shown in Fig. 7.5. Show that PMTPE is equivalent to the equilibrium conditions inside and at the ends of the beam.

Fig. 7.5

Solution. For a beam segment loaded as shown, the total potential energy is

( )

_{l l l l}

l l

v v v v

v

v x p x M ₀ M T ₀ T

I

E ′′ − + ′ − ′ − +

= 2

∫

^2d

∫

^{d 0 0}

Π 1 .

At equilibrium, Π is stationary, δΠ =0 or

( )

d δ d δ δ δ δ 0

δ ′′ − + ₀ ′ − ′ − ₀ + =

′′

∫

∫

^{l l l l}

l l

v v

v v

v v

v x p x M ₀ M T ₀ T

I

E . (a)

Integrating by parts the first term gives

7. ENERGY METHODS 143

Integrating by parts the last term gives

( ) ( ) ( ) ( ) ( )

The coefficient of δ in the integrand gives the equation of equilibrium v

(

EIv′′

)

″= p

( )

x .

As δ is arbitrary, the other terms deliver the equilibrium conditions at the v beam ends

(

EIv′′

)

0 =M0, or δv′₀ =0,

(

EIv′′

)

0′=T0, or δv₀ =0,

(

EIv′′

)

_l =M_l, or δv′_l =0, and

(

EIv′′

)

_l′ =T_l, or δv_l =0, which are the boundary conditions.

7.3 The Rayleigh-Ritz method

The Rayleigh-Ritz method involves the construction of an assumed displacement field. For a beam, the transverse displacement v

( )

x is approximated by a finite series

( ) ∑ ( )

=

≅

n

j

j

j x

a x

1

v ϕ (7.23)

where a are undetermined constants called generalized coordinates, and _j ϕ_j

( )

x are prescribed functions of x , called admissible functions, that satisfy the kinematic (geometric) boundary conditions and are continuous within the definition interval.

Substituting the displacements (7.23) into the expression of the total potential energy Π , the latter becomes a function of the parameters a , whose _j values are determined from the stationarity conditions

0 δ

δ =

∂

=

∑

∂

j

j j

a a

Π Π .

Because δ are arbitrary, a_j

=0

∂

∂ aj

Π ,

(

j=1,...,n

)

, (7.24) which is a linear algebraic set of equations in the constants a . _j

The solutions are back-substituted into (7.23) which represents an approximate deflected shape, which is more accurate the more terms are selected in the respective series.

The necessary requirements for the convergence of the Rayleigh-Ritz method are the following:

a) The approximating functions must be continuous to one order less the highest derivative in the integrand.

b) The functions must individually satisfy the geometric boundary conditions, i.e. to be admissible functions.

c) The sequence of functions must be complete.

If the functions are not selected from the domain space of the operator of the equation being solved (completeness property) the resulting solution could be either zero or wrong.

Example 7.4

For the beam shown in Fig. 7.6 find the vertical displacement of point 2.

Consider: MPaE=210 , I =1600mm⁴, ml=3 , NF=100 and q=200N m. Solution. The total potential energy is

7. ENERGY METHODS 145

The geometric boundary conditions are

( )

0 =0 In the following, for simplicity, the transverse displacements are approximated by a series consisting of only two terms

( )

^x =^a1ϕ1

( )

^x +^a2ϕ2

( )

^x

v , (7.27)

where the functions

( ) ( )( )

satisfy all geometrical boundary conditions (7.26).

Fig. 7.6

Substituting (7.27) into (7.25) we obtain the functional

( ) ( )

equations relating the generalized coordinates

( )

^{d d} 1

^{( )}

^{2 0}

( )

^{d d} 2

^{( )}

^{2 0}

For example, the coefficient of a in the first equation is ₁

( ) ( )

₃

The solutions are

I In 2, the displacement is

( ) ( ) ( )

Example 7.5

Consider a linear spring of stiffness k (Fig. 7.7, a) subjected to a load F.

Comment on the approximations of the Rayleigh-Ritz method.

Answer. The total potential energy (Fig. 7.7, b) is u

For a small virtual displacement uδ , the variation of the total potential energy is

7. ENERGY METHODS 147

The equilibrium equation is obtained by requiring δΠ to be zero (Π be stationary) for arbitrary uδ .

Fig. 7.7 For 0δΠ = we obtain

=0

− F u

k _eq . (7.31)

As seen in Fig. 7.7, b, the exact solution corresponds to an absolute minimum value of Π .

The total potential energy of the equilibrium configuration is

k u F

F u

F u

F _{eq eq eq}

eq

2

2 1 2

1 2

1 − =− =−

Π = . (7.32)

The stiffness is

eq eq

F k F

Π Π

2 2

2 1 2

1 =

−

= . (7.33)

The strain energy is

eq eq

eq k

u F k

U = ² = ² =−Π 2

1 2

1 . (7.34)

If F is prescribed and the resulting u has been approximated by a Rayleigh-Ritz solution u_app ≠u_eq, equations (7.32) – (7.34) and Fig. 7.7, b indicate that:

a) Because Π_eq is a minimum, the potential energy for an approximate displacement which satisfies the kinematic boundary conditions is greater than the true value

eq

app Π

Π > . In magnitude

eq

app Π

Π < ,

the approximate total potential energy is underestimated.

b) The approximate stiffness is overestimated

↑= ↓

eq

k F

Π

2

2

1 . (7.35)

c) The approximate displacement is underestimated

↓= ↑ k

u F . (7.36)

An approximate compatible displacement field corresponds to a structure which is stiffer than the actual structure and therefore will give a lower bound on displacement.

7.4 F.E.M. - a localized version of the Rayleigh-Ritz method

Instead of finding an admissible function satisfying the boundary conditions for the entire domain, which is often difficult, in the FEM the admissible functions are defined over small size subdomains.

7.4.1 F.E.M. in Structural Mechanics

a) Problem. Given a geometrically complex structure (including the boundary conditions) and the external loads

{ }

^{p ,} _v

{ }

^{p ,} _s

{ }

F , find the _i displacement field

{ }

u within V and on the surface S (Fig. 6.1). Then determine _σ stresses, internal forces, reaction forces, etc.

b) Solution approach. Use PVD or PMTPE as an approximate method for solving the boundary-value problem. Admissible functions are defined over small size finite elements, with simple geometry and well identified structural behaviour.

With these individually defined functions matching each other at certain points (nodes) at the element interfaces, the unknown function is approximated piecewise over the entire domain (continuity at global level).

c) Procedure. The geometric shape and the internal displacement field are described by a series of discrete quantities (like nodal coordinates and nodal displacements) distributed through the structure. For this a matrix notation is used.

d) Tools. Computers are used to store long lists of separate numbers and to manipulate them, to present output data in an engineering format, taking advantage of graphical and animation facilities.

7. ENERGY METHODS 149

7.4.2 Discretization

The structure is divided into finite elements (Fig. 7.8) that define the mesh.

Elements are defined by their nodal coordinates and some physical parameters.

Fig. 7.8

7.4.3 Principle of virtual displacements

For the entire structure, equation (7.6, a) can be written (considering only surface tractions)

{ } { }

δ d

{ } { }

δ d 0

δ =

∫

−

∫

=

σ

σ ε Π

S

T V

T V u p_s A . (7.37)

As a summation of virtual works on all elements, PVD yields

{ } { }

δ d

{ } { }

δ d 0 δ

δ =

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜

⎝

⎛

−

=

=

∑ ∑ ∫ ∫

e S

T V

T e

e

e e

A p u V

σ

σ ε Π

Π _s . (7.38)

In the following, only δΠ^e will be considered. The aim is the calculation of the element stiffness matrix and load vector.

7.4.4 Approximating functions for the element

In the Rayleigh-Ritz method, the trial function is expressed as a finite expansion

( ) ( ) _{⎣ ⎦ ⎣ ⎦} { }

^a

a a a x

a x

n n n

j

j

jϕ ϕ ϕ ϕ = ϕ

⎪⎪

⎭

⎪⎪

⎬

⎫

⎪⎪

⎩

⎪⎪

⎨

⎧

=

≅

∑

= L L²

1

2 1 1

u (7.39)

where the undetermined constants a have no direct evident signification. _j

The basic idea of FEM is to choose the constants - the displacement unknowns at the nodes

{ }

^{a =}

{ }

^Qe and to prescribe admissible functions denoted

⎣ ⎦ ⎣ ⎦

^{ϕ = N so that}

⎣ ⎦

⎣ ⎦

⎣ ⎦

{ } { } { }

^⎪_⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡

⎪=

⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

e e ue u

Q Q Q

N N N

w v w

v

w v u

0 0

0 0

0 0

or

{ }

^{u =}

[ ]

^N

{ }

^Qe , (7.40)

where

[ ]

N is the matrix of shape functions (interpolation functions).

The reason is that elements are small enough so that the shape of the displacement field can be approximated without too much error and only the magnitude, defined by

{ }

Q remains to be found. ^e

The proper selection of shape functions ensure the continuity of the displacement field at global level. A finite element described by admissible shape functions (integrable in the interior and with equal values of generalized coordinates at element interfaces) is referred to as co-deformable or conforming.

7.4.5 Compatibility between strains and nodal displacements

From the compatibility relationship (6.18)

{ }

^{ε =}

[ ]{ } [ ][ ]

^{∂ u = ∂ N}

{ }

^{Qe =}

^{[ ]}

^B

{ }

^{Qe , (7.41)}

where

[ ]

B is the matrix of differentiated shape functions.

The strain virtual variation is

{ }

^{δε =}

[ ]

^{B δ}

{ }

^{Qe .}

7. ENERGY METHODS 151

7.4.6 Element stiffness matrix and load vector

Using the constitutive equation

{ }

^{σ =}

[ ]{ }

^{D ε} , the virtual work for an element is

{ }

^δ

^{[ ] [ ][ ]} { }

^d

{ }

^δ

^{[ ] { }}

^{d 0}

δ =

∫

−

∫

=

e

e A

T T e V

e T T

e

e Q B D B Q V Q N p_s A

Π or

^δ

{ }

^δ

^{[ ] [ ][ ]}

^d

{ } ^{[ ] { }}

^d _⎟⎟^⎟=0

⎠

⎞

⎜⎜

⎜

⎝

⎛ −

=

∫ ∫

e

e A

T e

V T T e

e Q B D B V Q N p_s A

Π .

As

{ }

^δQe are arbitrary and non-zero, cancelation of the bracket yields the element equilibrium equation

[ ]

^Ke

{ } { }

^{Qe = Fe , (7.42)}

where the element stiffness matrix is

[ ]

⁼

∫ ^{[ ] [ ][ ]}

Ve T

e B D B V

K d (7.43)

and the vector of consistent nodal forces is

{ }

⁼

_∫ ^{[ ] { }}

Ae

e N T p A

F _s d . (7.44)

7.4.7 Assembly of the global stiffness matrix and load vector

In the next step, all individual elements are assembled together so that the displacements are continuous across element interfaces and the boundary conditions are satisfied.

The kinematic connectivity is expressed by the relationship between element and global displacements

{ }

^{Qe =}

[ ]

^T~e

{ }

^{Q , (7.45)}

where

{ }

^Qe is the vector of nodal element displacements,

{ }

Q is the vector of the global displacements of the structure and

[ ]

^T~e is a connectivity matrix, containing ones at the nodal displacements of element nodes and zeros elsewhere.

The variation of element displacements is

{ }

^{δQe =}

[ ]

^T~e

{ }

^{δQ . (7.46)}

The PVD equation for the entire structure is

{ } [ ] { } ∑ { } { }

∑

⁼

e

T e e e

e T e

e K Q Q F

Q δ

δ ,

or using (7.45) and (7.46)

{ } ∑ [ ] [ ][ ] { } { }

⁼

∑ [ ] { }

e

T e T e

e

e T e

T e

Q T~ Q

Q T~ K T~

Q δ

δ .

As

{ }

^δQ are arbitrary and non-zero, the unreduced global equilibrium equations are

[ ]

^K

{ } { }

^{Q = F , (7.47)}

where the global stiffness matrix is

[ ]

⁼

∑ [ ] [ ][ ]

e

e T e

e T~

K T~

K (7.48)

and the global load vector is

{ }

⁼

∑ [ ] { }

e

T e

e F

T

F ~

. (7.49) Applying the boundary conditions, the condensed equilibrium equations are

[ ]{ } { }

K Q = F . (7.50) The above procedure is never used in practice. It has been used only to show algebraically how to assemble a global stiffness matrix. The assembly is done by directly placing the nonzero entries of element stiffness matrices in the right locations of the global stiffness matrix based on element connectivity.

7.4.8 Solution and back-substitution

Nodal displacements are determined by solving the linear equations (7.50).

In the back-substitution phase, element stresses are evaluated as

{ }

^{σ =}

[ ]{ } [ ][ ]

^{D ε = D B}

{ }

^{Qe =}

^{[ ][ ]}

^{D B}

[ ]

^T~e

{ }

^Q

where

[ ]

D is given by (6.24) or (6.25).

8 .

In document FACULTAD DE CIENCIAS DE LA SALUD ESCUELA PROFESIONAL DE PSICOLOGÍA (página 33-41)